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Quadratic Expressions

Introduction

Quadratic Expressions are algebraic expressions in which the highest power of the variable is 2. A typical one-variable quadratic expression has the form $a x^{2} + b x + c$ , where $a$ , $b$ , and $c$ are coefficients and $a \neq 0$ . In Grades 9–10, learning to recognize, expand, factor, and rewrite quadratic expressions helps you understand polynomials, solve quadratic equations, and interpret quadratic functions.

The graph above shows the basic square function $y = x^{2}$ . Although this course focuses first on expressions, the same algebraic structure controls the shape and key features of a quadratic graph.

Learning Goals

By the end of this aiMOOC, you should be able to:

Recognize quadratic expressions: Identify the degree, variable, terms, and coefficients of a quadratic expression.
Expand products: Use the distributive property to multiply binomials and simplify like terms.
Factor quadratic expressions: Reverse expansion using common factors, factor pairs, special products, and grouping.
Complete the square: Rewrite a quadratic expression in a form that reveals a perfect square.
Connect forms and graphs: Explain what standard, factored, and vertex forms reveal.
Apply quadratics: Build and interpret quadratic expressions in geometric and real-world situations.

Understanding Quadratic Expressions

Terms, Coefficients, and Degree

Consider $3 x^{2} - 5 x + 7$ . It has three terms: the quadratic term $3 x^{2}$ , the linear term $- 5 x$ , and the constant term $7$ . The coefficient of $x^{2}$ is 3, the coefficient of $x$ is -5, and the constant term is 7. The degree is 2 because the greatest exponent of $x$ with a nonzero coefficient is 2.

The requirement $a \neq 0$ matters. If the coefficient of $x^{2}$ were zero, then $a x^{2} + b x + c$ would no longer be quadratic.

The image compares families of quadratic graphs while the coefficients vary. It helps you see that changing algebraic coefficients changes graphical behavior.

Equivalent Expressions

Two expressions are equivalent if they have the same value for every allowed value of the variable. For example,

$(x + 2) (x + 5) = x^{2} + 7 x + 10$ .

The left side is in factored form and the right side is in standard form. You can verify equivalence by expanding: $x (x + 5) + 2 (x + 5) = x^{2} + 5 x + 2 x + 10 = x^{2} + 7 x + 10$ .

A useful habit is to check an algebraic rewrite in two ways: first by applying algebraic rules, and then by substituting one or two simple values such as $x = 0$ or $x = 1$ . Numerical checks cannot prove equivalence by themselves, but they can reveal many mistakes.

Expanding Quadratic Expressions

Multiplying Two Binomials

To expand $(x + p) (x + q)$ , distribute every term in the first binomial across every term in the second:

$(x + p) (x + q) = x^{2} + (p + q) x + p q$ .

For example,

$(x + 4) (x - 3) = x^{2} + x - 12$ .

The middle coefficient comes from the sum $4 + (- 3) = 1$ , while the constant comes from the product $4 (- 3) = - 12$ .

When the leading coefficients are not both 1, the same distributive idea applies:

$(2 x + 3) (x - 4) = 2 x^{2} - 8 x + 3 x - 12 = 2 x^{2} - 5 x - 12$ .

Special Products

Some patterns are worth recognizing because they make expansion and factorization faster:

$(x + p)^{2} = x^{2} + 2 p x + p^{2}$

$(x - p)^{2} = x^{2} - 2 p x + p^{2}$

$(x + p) (x - p) = x^{2} - p^{2}$

The first two are perfect-square trinomials. The third is the difference of squares. These are not tricks; each pattern follows from the distributive property.

Factoring Quadratic Expressions

Factoring reverses expansion. You rewrite a sum or difference as a product. Always look first for a greatest common factor. For example, $6 x^{2} + 9 x = 3 x (2 x + 3)$ .

Factoring When the Leading Coefficient Is 1

For $x^{2} + b x + c$ , look for two numbers whose sum is $b$ and whose product is $c$ . For example, to factor $x^{2} + 7 x + 12$ , the numbers 3 and 4 work because $3 + 4 = 7$ and $3 \cdot 4 = 12$ . Therefore,

$x^{2} + 7 x + 12 = (x + 3) (x + 4)$ .

The video gives a worked introduction to factoring quadratics of the form $x^{2} + b x + c$ . Pause before each worked step and try to predict the needed factor pair.

Factoring When the Leading Coefficient Is Not 1

For an expression such as $2 x^{2} + 7 x + 3$ , you can use grouping. Multiply the leading coefficient and the constant: $2 \cdot 3 = 6$ . Find two numbers that multiply to 6 and add to 7: 6 and 1. Split the middle term:

$2 x^{2} + 6 x + x + 3$

and group:

$2 x (x + 3) + 1 (x + 3) = (2 x + 1) (x + 3)$ .

After factoring, expand your result to check it.

Special Factoring Patterns

The difference of squares factors as $A^{2} - B^{2} = (A - B) (A + B)$ . For example, $x^{2} - 25 = (x - 5) (x + 5)$ .

A perfect-square trinomial such as $x^{2} + 10 x + 25$ factors as $(x + 5)^{2}$ . Check the middle term: twice the product of $x$ and 5 is $10 x$ .

Completing the Square

Completing the square rewrites a quadratic expression so that a perfect square is visible. For a monic quadratic,

$x^{2} + b x + c = {(x + \frac{b}{2})}^{2} + c - {(\frac{b}{2})}^{2}$ .

For example,

$x^{2} + 6 x + 2 = (x + 3)^{2} - 7$ .

You obtain this by adding and subtracting 9, because half of 6 is 3 and $3^{2} = 9$ .

Completing the square is useful because it connects symbolic manipulation to the vertex of a parabola and also provides a route to the quadratic formula.

Completing the Square with a Leading Coefficient

For $a x^{2} + b x + c$ , factor $a$ from the quadratic and linear terms before completing the square. In general,

$a x^{2} + b x + c = a {(x + \frac{b}{2 a})}^{2} + c - \frac{b^{2}}{4 a}$ .

For example,

$2 x^{2} + 8 x + 3 = 2 (x^{2} + 4 x) + 3 = 2 (x + 2)^{2} - 5$ .

This equivalent form makes the minimum value of the related function easy to identify when $a > 0$ .

Three Useful Forms

A quadratic expression can often be written in several equivalent forms. Each form highlights different information.

Standard form: $a x^{2} + b x + c$ makes the coefficients and y-intercept of the related function easy to see.
Factored form: $a (x - r_{1}) (x - r_{2})$ makes real zeros visible when such a factorization exists.
Vertex form: $a (x - h)^{2} + k$ makes the vertex $(h, k)$ of the related function visible.

The graph connects algebraic features with zeros, the vertex, the axis of symmetry, and the y-intercept. Switching forms is therefore not merely symbolic practice; it is a way to reveal different mathematical information.

From Expressions to Equations

An expression such as $x^{2} - 5 x + 6$ has no equality sign. If you set it equal to zero, you create the quadratic equation $x^{2} - 5 x + 6 = 0$ . Because $x^{2} - 5 x + 6 = (x - 2) (x - 3)$ , the zero-product property gives solutions $x = 2$ and $x = 3$ .

If a quadratic cannot be factored conveniently, completing the square or the quadratic formula can be used. For $a x^{2} + b x + c = 0$ with $a \neq 0$ ,

$x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$ .

The quantity $b^{2} - 4 a c$ is the discriminant. For real coefficients, a positive discriminant gives two distinct real roots, zero gives one repeated real root, and a negative discriminant gives no real roots.

For this course, the important connection is structural: factoring an expression, completing the square, and using the quadratic formula are different ways of revealing information hidden inside the same quadratic coefficients.

Applications and Modelling

Area Models

Quadratic expressions appear naturally in area. A rectangle with side lengths $x + 3$ and $x + 5$ has area

$(x + 3) (x + 5) = x^{2} + 8 x + 15$ .

The factored form describes the side lengths; the expanded form describes the total area as a sum of component areas. Drawing an area model can make the distributive property visible.

This animation illustrates a geometric relationship between the square function and equal-area square and rectangle constructions, reinforcing the connection between quadratic algebra and area.

Motion and Maximum or Minimum Values

Under a simple constant-gravity model with air resistance ignored, the height of a moving object can be represented by a quadratic function of time. In SI units, a model may take the form $h (t) = - 4.9 t^{2} + v_{0} t + h_{0}$ , where $v_{0}$ is initial vertical velocity and $h_{0}$ is initial height. Completing the square or finding the vertex can reveal the model's maximum height.

A model is always an approximation. You should state assumptions, choose meaningful units, and decide whether the values predicted by the expression make sense in the situation.

Common Errors and How to Check Them

Incomplete distribution: In $(x + 3) (x + 4)$ , every term in one factor must multiply every term in the other.
Combining unlike terms: $x^{2}$ and $x$ are not like terms and cannot be added into one term.
Sign errors in factoring: Check both the sum and product of your chosen factor pair.
Incorrect perfect-square pattern: Remember that $(x + p)^{2}$ includes the middle term $2 p x$ .
Changing an expression: When you add a value to create a square, subtract the same value in the same expression unless you are working with an equation and balancing both sides.
Skipping a check: Re-expand a factored form or compare values to catch arithmetic mistakes.

Interactive Tasks

Quiz: Test Your Knowledge

What makes an expression quadratic in one variable? (The highest nonzero power of the variable is 2) (!The expression contains exactly two terms) (!The variable has coefficient 2) (!The constant term is 2)

What is the leading coefficient of 3x squared minus 5x plus 7? (3) (!Minus 5) (!7) (!2)

What is the expansion of x plus 4 times x minus 1? (x squared plus 3x minus 4) (!x squared plus 5x minus 4) (!x squared minus 3x minus 4) (!x squared plus 3x plus 4)

Which text gives the correct factorization of x squared plus 7x plus 12? (x plus 3 times x plus 4) (!x plus 2 times x plus 6) (!x minus 3 times x minus 4) (!x plus 1 times x plus 12)

Which expression is a perfect-square trinomial? (x squared plus 6x plus 9) (!x squared plus 6x plus 8) (!x squared plus 9x plus 6) (!x squared minus 6x minus 9)

What number completes the square for x squared plus 8x? (16) (!4) (!8) (!64)

What information is especially visible in factored form? (The zeros of the related quadratic function) (!Only the y intercept) (!Only the leading coefficient) (!The domain of every function)

What are the zeros of x squared minus 5x plus 6? (2 and 3) (!Minus 2 and minus 3) (!1 and 6) (!Minus 1 and minus 6)

What does a negative discriminant mean for a quadratic with real coefficients? (There are no real roots) (!There are exactly two equal real roots) (!There are always two positive roots) (!The expression is linear)

Why is re-expanding a factorization useful? (It checks whether the factorization is equivalent) (!It changes the degree to 1) (!It removes the constant term) (!It guarantees integer roots)

Memory Game

Quadratic	An expression whose highest nonzero variable power is two
Coefficient	A numerical multiplier of a variable term
Factorization	Rewriting an expression as a product
Discriminant	The value that classifies the real roots of a quadratic equation
Vertex	The maximum or minimum turning point of a parabola
Expansion	Multiplying factors and combining like terms

Drag and Drop

Match the correct terms.	Topic
Standard form	Displays the quadratic, linear, and constant coefficients directly
Factored form	Reveals zeros of the related function when real linear factors are present
Vertex form	Reveals the turning point of the related parabola
Common factor	Should be checked before other factoring methods
Completing the square	Rewrites a quadratic by creating a perfect-square expression

...

Crossword Puzzle

Quadratic	What word describes a polynomial expression of degree two?
Coefficient	What is the numerical multiplier attached to a variable term?
Factor	What word names an expression multiplied by another expression in a product?
Parabola	What curve is the graph of a one-variable quadratic function?
Vertex	What is the turning point of a parabola called?
Discriminant	What quantity b squared minus 4ac classifies the real roots of a quadratic equation?

LearningApps

Cloze Text

Open-Ended Tasks

Easy

Expression Spotter: Collect ten algebraic expressions from a textbook or create your own, classify each as quadratic or not quadratic, and explain your decisions in one sentence each.
Area Model Poster: Draw an area model for $(x + 3) (x + 5)$ , label every region, and use the diagram to explain why the product equals $x^{2} + 8 x + 15$ .
Expansion Tutorial: Record a short video or screen capture teaching how to expand one pair of binomials, including a final substitution check.
Factor-Pair Hunt: Create a set of cards for six quadratics of the form $x^{2} + b x + c$ and challenge a partner to match each expression with the pair of numbers whose sum is $b$ and product is $c$ .

Standard

Real-World Interview: Interview a teacher, engineer, designer, builder, or another adult about a situation involving area, optimization, or curved motion, then explain where a quadratic expression could appear in the mathematics.
Graphing Lab: Use graphing technology to compare $y = a x^{2}$ for at least five nonzero values of $a$ , record what changes, and connect your observations to the coefficient.
Completing-the-Square Explanation: Produce an annotated one-page explanation showing how to rewrite $x^{2} + 8 x + 3$ in vertex form and explain why each step preserves equivalence.
Quadratic Shape Gallery: Visit your school or community, photograph or sketch at least three curved objects that appear approximately parabolic, and explain why visual resemblance alone does not prove an exact quadratic model.

Advanced

Motion Modelling Experiment: Safely toss a soft ball, use video frames to estimate height at several times, fit a quadratic model with suitable software, and discuss measurement error and model limitations.
Method Comparison: Choose one quadratic equation and solve it by factoring when possible, completing the square, and the quadratic formula; compare efficiency and explain what each method reveals.
Error-Analysis Podcast: Create a three- to five-minute audio or video explanation of three common quadratic-expression mistakes, using incorrect examples, corrected work, and checking strategies.
Design Challenge: Investigate a rectangle with a fixed perimeter, build a quadratic expression for its area, determine the maximum area, and present both algebraic and graphical justification.

Learning Assessment

Structure and choice: Given six quadratic expressions in different forms, identify what information each form reveals and justify which form you would choose for expansion, roots, or a vertex.
Equivalence argument: Prove algebraically that $2 (x + 2)^{2} - 5$ and $2 x^{2} + 8 x + 3$ are equivalent, then explain why testing only one numerical value would not be a proof.
Method selection: For three quadratic equations, choose between factoring, completing the square, and the quadratic formula; solve each and defend your choice based on structure.
Model interpretation: Build a quadratic expression from an area or motion context, define every variable and unit, and explain which parts of the expression carry contextual meaning.
Critique and correction: Analyze a worked solution containing a sign error and an incomplete distribution, locate each error, correct the work, and describe a check that would have detected it.
Transfer across representations: Starting from a quadratic in standard form, rewrite it in another useful form, sketch the corresponding graph, and explain how the algebra predicts at least two graphical features.

Evidence of Learning

Knowledge: You accurately use terms such as degree, coefficient, factor, perfect square, vertex, zero, and discriminant.
Skills: You expand, simplify, factor, complete the square, check equivalence, and move between useful forms of quadratic expressions.
Reasoning: You justify algebraic transformations and choose methods based on the structure of an expression rather than following one fixed procedure.
Products: Your portfolio can include annotated solutions, area models, graphs, a short tutorial, a modelling report, and an error analysis.
Communication: You explain steps in clear mathematical language, define variables and units, and distinguish an expression from an equation or function.
Transfer: You apply quadratic structure to unfamiliar geometry, motion, optimization, and graph interpretation problems while stating assumptions and checking whether results are reasonable.

OERs on the Topic

The English Wikipedia article on quadratic functions provides a useful open reference for quadratic polynomials, their forms, graphs, coefficients, and related equations.

Linked Learning Areas

Quadratic Expressions

Quadratic expressions connect algebra with geometry, functions, graphing, motion models, and optimization. These connections help you move between symbolic, visual, numerical, and contextual representations.

aiMOOC Projects